The text itself is not as specific as some might like. I would say that for the meaning of the verse, something like "the faith that he has" or "his faith", since the first part of the verse talks about someone having faith. The second part literally says only "the faith" but it doesn't mean "the faith" like we would use today, as in the Christian faith in general... it is talking specifically of that person's faith (in my opinion).

Now for my next question: Does the second part refer to his claimed faith, i.e. the faith that he claims to have but doesn't actually have or to his faith that he actually has?

It's hard to say because the text is not that specific. It is a judgment call of the reader.

My impression is that it is referring to the faith that he claims to have... he may or may not have any faith but he is *saying* he has faith, but is not following through, and so *that* faith cannot save him.